My colleague asked the following question as part of our daily brain teaser but we have differing answers. So any help would be great for finding the “correct” answer.

If all of the world’s population of 7.78bn people followed social distancing rules and stood 6 feet apart, what’s the smallest country in the world they would fit in to?

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    $\begingroup$ What is the optimum packing for circles? $\endgroup$ May 27 '20 at 8:27
  • $\begingroup$ I used triangles in my attempt. $\endgroup$
    – novice
    May 27 '20 at 8:50
  • $\begingroup$ Correct, equilateral triangles are how you should map it. $\endgroup$ May 27 '20 at 8:53
  • $\begingroup$ You mean if all the population stood 2 meters apart ? :) $\endgroup$
    – Jean Marie
    May 27 '20 at 17:46
  • $\begingroup$ Optimum packing is an equlateral triangle grid. Figure out the area of each these triangles (are we asssuming people are points and the sides are $6$ feet or that a person has body radius and we must add that to the six feet). Then... just do it. $7.78$ billion of these triangles is so and so area and then get a list of countries by land area.... $\endgroup$
    – fleablood
    May 27 '20 at 18:23

While the problem itself is very simple, one can be easily trapped in an argument between equilateral triangles and regular hexagons; but that is an illusion: people in a triangular grid, in hexagonal cells Both of these tessellate the 2D plane, with the same density of people (red dots).

The key is to calculate the area needed per person correctly.

On one hand, we can say that each person stands at a grid point of a regular equilateral triangle grid, with grid spacing (and equilateral triangle edge length) the same as the interperson distance $d = 6\text{ ft} = 1.8288\text{m }$. Each person is at the vertex shared by six triangles, but there are just three vertices per triangle; so, the density of people is three times one sixth of a person per triangle, or half a person per triangle. In other words, each person uses up the area of two triangles: $$A_\Delta = 2 \frac{\sqrt{3}}{4} d^2 = \frac{\sqrt{3}}{2} d^2 \approx 2.8964\text{ m}^2$$ or about $31.177\text{ sq ft}$.

On the other hand, we can say that each person stands inside a regular hexagon with edge length $a$. Consider the yellow right angle part in the above diagram: the horizontal part is $d/2$, vertical part is $a/2$, and the angle at the person (red dot) is $30°$; with $d = 6\text{ ft} = 1.8288\text{m }$ the interperson distance. In other words, $$\frac{a / 2}{d / 2} = \tan 30° \quad \iff \quad a = \frac{d}{\sqrt{3}}$$ and the area each person occupies is $$A_⬡ = \frac{3\sqrt{3}}{2} a^2 = \frac{3\sqrt{3}}{2}\left(\frac{d}{\sqrt{3}}\right)^2 = \frac{\sqrt{3}}{2} d^2 \approx 2.8964\text{ m}^2$$ or about $31.177\text{ sq ft}$.

So, saying that people stand at grid points of a regular equilateral triangular grid, or that people occupy hexagonal cells or tiles, is essentially just two different ways to say the same thing; either way, each person will occupy an area of $$A_{1} \approx 2.8964\text{ m}^2 \approx 31.177\text{ sq ft}$$ when tessellating the 2D plane (packing them on a flat plane).

This means that the area needed for roughly $7\,780\,000\,000$ people is approximately $$A_\text{total} \approx 22\,534\text{ km}^2 \approx 8\,701\text{ sq mi}$$

As to which country is the smallest but has at least $22\,534\text{ km}^2$ surface people could stand on – including actual area along hillsides, whether to consider only land area or include inland lakes that freeze hard enough during winters to stand on – is not really a geometry question, but a geographical or perhaps political question; we humans tend to disagree even which areas are their own countries, or just parts of other countries.. so I won't be making any statements or claims on that, I'm here just for the math and geometry part, and to show how to estimate the surface area needed.

If one of your friends calculates the area needed based on circles with radius $r = d/2$, i.e. half the interpersonal distance, they'll arrive an answer of $20\,436\text{ km}^2$ or $7\,890\text{ sq mi}$. The problem with that is that a circle does not tessellate a plane; there is always a small area left uncovered between three circles touching each other, that is not included in that area but is needed for the circles to not overlap. If you were to calculate or estimate the area of those uncovered parts, you'd arrive at the difference, or about $2\,098\text{ km}^2$ or $811\text{ sq mi}$.

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    $\begingroup$ Going on your answer, Wikipedia suggests that the answer is Belize, with El Salvador being the largest country that's too small. $\endgroup$ May 27 '20 at 18:22
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    $\begingroup$ I'd quibble that people aren't points and that keeping the surface of the human body six feet for the surface of the next human adds $2$ feet to the sides I'd replace $6$ with $8$ and we'd require a country that is $(\frac {8}6)^2= 1\frac 79 \approx 1.77$ times bigger $\endgroup$
    – fleablood
    May 27 '20 at 18:29
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    $\begingroup$ @fleablood Switzerland or Estonia then. $\endgroup$ May 27 '20 at 18:36
  • $\begingroup$ @fleabood: ($\approx 1.78$) If you use $8\text{ ft} = 2.4384\text{ m}$ as the distance between the centers of each person, the area per person $A_1 \approx 5.1492\text{ m}^2$, and the total area needed is $A_\text{total} \approx 40\,061\text{ km}^2$ or about $15\,467\text{ sq mi}$. With this, if I were to guess (so this is not part of my answer), I'd say Estonia. Switzerland is too mountainous: we'd have to take into account that on a steep hillside, peoples head would be close to others' feet and vice versa, increasing the distance needed by several feet. Fun to think about, though. $\endgroup$
    – None
    May 27 '20 at 19:28
  • $\begingroup$ Thanks, this answer was what I was looking for confirming and it seems I had calculated right :) the question wasn’t heavy enough to go into the geography of the country and how much land is actually stand-able or the size of a human just a light morning question between a group of rusty mathematics grads and mathematically minded colleagues. BUT that being said those are interesting points that would be cool to include to make it much more challenging! $\endgroup$
    – novice
    May 28 '20 at 23:51

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